Posterior consistency for scale-mixture shrinkage priors in low dimension?

Question

Consider the model [1]

$$y_n=X_n\beta_n+\epsilon_n$$ $$\beta_i|\sigma^2,v_i \sim \mathcal{N}(0,\sigma^2 v_i), i=1,\ldots,p$$ $$v_i \sim \beta^\prime(a,b)$$ $$\sigma^2 \sim \mathcal{IG}(c,d)$$

where $\beta^\prime$ is the beta prime distribution and $\mathcal{IG}$ the inverse gamma distribution.

In [1], the authors prove posterior consistency in the high-dimensionality regimes were $p\rightarrow \infty$ as $n\rightarrow \infty$. Is there a way to show posterior consistency for fixed/small $p$ as $n\rightarrow \infty$?

[1] Bai, R. and Ghosh, M., 2021. On the beta prime prior for scale parameters in high-dimensional bayesian regression models. Statistica Sinica, 31(2), pp.843-865.

Answer 1

Based on your reference I believe that you are estimating the vector $\boldsymbol{\beta}$ of size $p_n$ with a posterior distribution based on the observation of the vector $\mathbf{Y}$ of size $n$ in the model $$\mathbf{Y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\epsilon}$$ where $\mathbf{X}$ is a fixed regression matrix $n \times p_n$ matrix. The components $\beta_i$ from the vector $\boldsymbol{\beta}$ have a prior distribution that follows the model that you described.

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If you have a fixed $p$ then consistency seems guaranteed by a Bayesian law of large numbers, or consistency of likelihood/posterior for an i.i.d sample (When do posteriors converge to a point mass?). Or maybe I am missing something?

Say we use some fixed $p$ while letting $n$ increase, then the Bayesian posterior density $f(\boldsymbol{\beta}|\mathbf{Y})$ will concentrate near the true $\boldsymbol{\beta}$ (if the prior is not zero).

The same is true when $p$ is not fixed, but still with a finite upper bound. We can consider all $p$'s below the bound together while letting $n \to \infty$ and if they are all individually consistent, then we will also have consistency when we change $p$ while letting $n$ increase.